Convergence of Rothe scheme for hemivariational inequalities of parabolic type

This article presents the convergence analysis of a sequence of piecewise constant and piecewise linear functions obtained by the Rothe method to the solution of the first order evolution partial differential inclusion $u'(t)+Au(t)+\iota^*\partial J(\iota u(t))\ni f(t)$, where the multivalued term is given by the Clarke subdifferential of a locally Lipschitz functional. The method provides the proof of existence of solutions alternative to the ones known in literature and together with any method for underlying elliptic problem, can serve as the effective tool to approximate the solution numerically. Presented approach puts into the unified framework known results for multivalued nonmonotone source term and boundary conditions, and generalizes them to the case where the multivalued term is defined on the arbitrary reflexive Banach space as long as appropriate conditions are satisfied. In addition the results on improved convergence as well as the numerical examples are presented.Comment: to appear in: International Journal of Numerical Analysis and Modelin

Global attractors for multivalued semiflows with weak continuity properties

A method is proposed to deal with some multivalued semiflows with weak continuity properties. An application to the reaction-diffusion problems with nonmonotone multivalued semilinear boundary condition and nonmonotone multivalued semilinear source term is presented.Comment: to appear in Nonlinear Analysis Series A, Theory, Methods & Application

Modified Einstein's gravity to probe the sub- and super-Chandrasekhar limiting mass white dwarfs: a new perspective to unify under- and over-luminous type Ia supernovae

Type Ia supernovae (SNeIa), used as one of the standard candles in astrophysics, are believed to form when the mass of the white dwarf approaches Chandrasekhar mass limit. However, observations in last few decades detected some peculiar SNeIa, which are predicted to be originating from white dwarfs of mass much less than the Chandrasekhar mass limit or much higher than it. Although the unification of these two sub-classes of SNeIa was attempted earlier by our group, in this work, we, for the first time, explain this phenomenon in terms of just one property of the white dwarf which is its central density. Thereby we do not vary the fundamental parameters of the underlying gravity model in the contrary to the earlier attempt. We effectively consider higher order corrections to the Starobinsky-$f(R)$ gravity model to reveal the unification. We show that the limiting mass of a white dwarf is $\sim M_\odot$ for central density $\rho_c \sim 1.4\times10^8$ g/cc, while it is $\sim 2.8M_\odot$ for $\rho_c \sim1.6\times 10^{10}$ g/cc under the same model parameters. We further confirm that these models are viable with respect to the solar system test. This perhaps enlightens very strongly the long standing puzzle lying with the predicted variation of progenitor mass in SNeIa.Comment: 15 pages including 6 figures: published in JCA

On non-autonomously forced Burgers equation with periodic and Dirichlet boundary conditions

We study the non-autonomously forced Burgers equation $$u_t(x,t) + u(x,t)u_x(x,t) - u_{xx}(x,t) = f(x,t)$$ on the space interval $(0,1)$ with two sets of the boundary conditions: the Dirichlet and periodic ones. For both situations we prove that there exists the unique $H^1$ bounded trajectory of this equation defined for all $t\in \mathbb{R}$. Moreover we demonstrate that this trajectory attracts all trajectories both in pullback and forward sense. We also prove that for the Dirichlet case this attraction is exponential

Attractors for Navier-Stokes flows with multivalued and nonmonotone subdifferential boundary conditions

We consider two-dimensional nonstationary Navier-Stokes shear flow with multivalued and nonmonotone boundary conditions on a part of the boundary of the flow domain. We prove the existence of global in time solutions of the considered problem which is governed by a partial differential inclusion with a multivalued term in the form of Clarke subdifferential. Then we prove the existence of a trajectory attractor and a weak global attractor for the associated multivalued semiflow. This research is motivated by control problems for fluid flows in domains with semipermeable walls and membranes.Comment: A correction was introduced in assertion (ii) of Definition 4.4 and - accordingly - in the proof of Theorem 4.